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5.11. Theorem. (Cauchy-Riemann operators on the strip) Let p ∈]1, +∞[, and assume that
Φ − (1)λ(−∞) ∩ ν(−∞) = (0), Φ + (1)λ(+∞) ∩ ν(+∞) = (0).
(i) The boundedÊ-linear operator
is Fredholm of index
∂A : W 1,p
λ,ν (Σ,�n ) → L p (Σ,�n ), ∂Au = ∂u + Au,
ind∂A = µ(Φ − λ(−∞), ν(−∞)) − µ(Φ + λ(+∞), ν(+∞)) + µ(λ, ν).
(ii) If furthermore A(s, t) = A(t), λ(s) = λ, and ν(s) = ν do not depend on s, the operator ∂A
is an isomorphism.
Note that under the assumptions of (ii) above, the equation ∂u + Au can be rewritten as
∂su = −LAu, where LA is the unboundedÊ-linear operator on L 2 (]0, 1[,�n ) defined by
domLA = W 1,2
λ,ν (]0, 1[,�n � � 1,2
) = u ∈ W (]0, 1[,�n
) | u(0) ∈ λ, u(1) ∈ ν , LA = i d
+ A.
dt
The conditions on A imply that LA is self-adjoint and invertible. These facts lead to the following:
5.12. Proposition. Assume that A, λ, and ν satisfy the conditions of Theorem 5.11 (ii), and
set δ := minσ(LA) ∩ [0, +∞[> 0. Then for every k ∈Æthere exists ck such that
�u(s, ·)� C k ([0,1]) ≤ ck�u(0, ·)� L 2 (]0,1[)e −δs , ∀s ≥ 0,
for every u ∈ W 1,p (]0, +∞[×]0, 1[,�n ), p > 1, such that u(s, 0) ∈ λ, u(s, 1) ∈ ν for every s ≥ 0,
and ∂u + Au = 0.
Next we need the following easy consequence of the Sobolev embedding theorem:
5.13. Proposition. Let s > 0 and let χs be the characteristic function of the set {z ∈ Σ | |Re z| ≤ s}.
Then the linear operator
X 1,p
S (Σ,�n ) → X q
S (Σ,�n ), u ↦→ χsu,
is compact for every q < ∞ if p ≥ 2, and for every q < 2p/(2 − p) if 1 ≤ p < 2.
Proof. Let (uh) be a bounded sequence in X 1,p
S (Σ,�n ). Let {ψ1, ψ2} ∪ {ϕj} k+k′
j=1 be a smooth
partition of unity of�satisfying (95). Then the sequences (ψ1uh), (ψ2uh) and (ϕjuh), for 1 ≤
j ≤ k + k ′ are bounded in X1,p (Σ,�n ). We must show that each of these sequences is compact in
X q
S (Σ,�n ).
Since the Xq and X1,p norms on the space of maps supported in Σ \ Br/2(S ) are equivalent
to the Lq and W 1,p norms, the Sobolev embedding theorem implies that the sequences (χsψ1uh)
and (χsψ2uh) are compact in X q
S (Σ,�n ).
Let 1 ≤ j ≤ k. If u is supported in Br(sj), set v(z) := u(sj + z), so that by (87)
�u� q
Xq (Σ) =
�
|u(z)| q |z| q/2−1 �
dsdt ≤
Br(sj)∩Σ
Br(sj)∩Σ
1
|z| |u(z)|q dsdt = 4�Ru� q
Lq (À+ ∩�√
r ) . (96)
Set vh(z) := ϕj(sj + z)uh(sj + z). By (91), the sequence (Rvh) is bounded in W 1,p (À+
∩�√
r),
hence it is compact in Lq (À+
∩�√
r) for every q < ∞ if p ≥ 2, and for every q < 2p/(2 − p) if
1 ≤ p < 2. Then (96) implies that (ϕjuh) is compact in X q
S (Σ,�n ). A fortiori, so is (χsϕjuh).
The same argument applies to j ≥ k + 1, concluding the proof.
Putting together Lemma 5.10, statement (ii) in Theorem 5.11, and the Proposition above we
obtain the following:
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